We construct combinatorial volume forms of hyperbolic three manifolds fibering over the circle. These forms define non-trivial classes in bounded cohomology. After introducing a new seminorm on exact bounded cohomology, we use these combinatorial classes to show that, in degree 3, the zero norm subspace of the bounded cohomology of an acylindrically hyperbolic group is infinite dimensional. In an appendix we use the same techniques to give a cohomological proof of a lower bound, originally due to Brock, on the volume of the mapping torus of a cobounded pseudo-Anosov homeomorphism of a closed surface in terms of its Teichmüller translation distance.
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Federico Franceschini, Roberto Frigerio, Maria Beatrice B. Pozzetti, Alessandro Sisto, The zero norm subspace of bounded cohomology of acylindrically hyperbolic groups. Comment. Math. Helv. 94 (2019), no. 1, pp. 89–139DOI 10.4171/CMH/456